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In mathematics, the second partial derivative test is a method in multivariable calculus used to determine if a critical point of a function is a local minimum, maximum or saddle point. Functions of two variables
Statistical analyses of multivariate data often involve exploratory studies of the way in which the variables change in relation to one another and this may be followed up by explicit statistical models involving the covariance matrix of the variables. Thus the estimation of covariance matrices directly from observational data plays two roles:
The probability content of the multivariate normal in a quadratic domain defined by () = ′ + ′ + > (where is a matrix, is a vector, and is a scalar), which is relevant for Bayesian classification/decision theory using Gaussian discriminant analysis, is given by the generalized chi-squared distribution. [17]
Multivariable calculus is used in many fields of natural and social science and engineering to model and study high-dimensional systems that exhibit deterministic behavior. In economics , for example, consumer choice over a variety of goods, and producer choice over various inputs to use and outputs to produce, are modeled with multivariate ...
Multiple Discriminant Analysis (MDA) is a multivariate dimensionality reduction technique. It has been used to predict signals as diverse as neural memory traces and corporate failure. [1] MDA is not directly used to perform classification. It merely supports classification by yielding a compressed signal amenable to classification.
It is closely related to Hotelling's T-square distribution used for multivariate statistical testing and Fisher's linear discriminant analysis that is used for supervised classification. [ 13 ] In order to use the Mahalanobis distance to classify a test point as belonging to one of N classes, one first estimates the covariance matrix of each ...
In characteristic different from 2, [10] the discriminant or determinant of Q is the determinant of A. [11] The Hessian determinant of Q is times its discriminant. The multivariate resultant of the partial derivatives of Q is equal to its Hessian determinant. So, the discriminant of a quadratic form is a special case of the above general ...
The determinant of the Hessian at is called, in some contexts, a discriminant. If this determinant is zero then x {\displaystyle \mathbf {x} } is called a degenerate critical point of f , {\displaystyle f,} or a non-Morse critical point of f . {\displaystyle f.}